Abstract

VILCHEZ LOBATO, Ma. Luisa; VELASCO MORENTE, Francisco; GONZALEZ ABRIL, Luis  and  ORTEGA, Juan Antonio. Hopf Bifurcations: Qualitative Analysis and Application to a Bioeconomic Model of Fisheries. Comp. y Sist. [online]. 2003, vol.6, n.4, pp.273-283. ISSN 2007-9737.

This paper deals with the theory of codimension one bifurcations applied to a dynamical system derived from applying Pontryagin's maximum principle to an optimal control problem for economic fisheries management Changes in the number and/or the stability of stationary states as well as the appearance of closed orbits, are analyzed. Some necessajy and/or sufficient conditions are established for the existence of Hopf bifurcations, according to the biological and economic parameters of the model. We focus our attention on the resulting limit cycles which can be candidates for solutions of the optimal control problem. The method is applied to the South Atlantic "chamelea gallina" fishery. To obtain the bifurcation values, we have made numerical computations by the use of Content and Vensim software. So, we make a qualitative description of the trajectories behaviour around the stationary states.

Keywords : Stability; Bifurcations; Hopf Theorem; Limit Cycle; Optimal Control.